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“Footnotes”

Since I have been called “obscure” (or worse), I try to provide amplifying information where it may be needed. Just like in your old school reports, many times I’ll do this by putting a little elevated symbol (superscript) at the end of a thought. Instead of looking at the bottom of the page for a matching number, now you can just click on the symbol and be automatically transported to the appropriate reference material. Since I no longer need numbers, I’ve replaced them with the following: “A” means you will be sent to an article on the Web confirming my remark, “B” will be a previous blog post, “C” sends you to one of my side comments, “D” means a definition, and “E” means something providing extra information. These symbols may be followed by a number if I need to refer to the reference again later.

I could also use inline notes (click on the gray-highlighted line) or traditional hyperlinks (click on the underlined words).

Tag: Blue-eyed monks

I recently ran across the Problem of the Blue-spotted Monks again at https://richardwiseman.wordpress.com/2011/04/04/answer-to-the-friday-puzzle-98/. Actually, this is a slight variation of the well-known Blue-Eyed Monk problem that can be found at https://en.wikipedia.org/wiki/Common_knowledge_%28logic%29, among other places. One site even called this problem “The Hardest Logic Puzzle in the World”A, but that was probably just a case of self-promotion. For today’s discussion, I chose the first version of the problem because it is simpler (we don’t have to worry about the case in which one person has red eyes). If you haven’t done so, go ahead and read the problem. We will discuss the solution in the next section.

Start With Induction

The classic answer uses mathematical inductionD to first consider the (trivial) case in which only one monk has the disease. Then they move on to the case of two monks. Their error begins in the case of three infected monks, assuming that the monks were required to start back at one in making their individual analysis. That is wrong; in the comments section of some of the references, several people take issue with this assumption. I believe this to be a misapplication of the induction process. The monks have only to consider two possibilities; either they are infected or they are not. For the sake of argument, let’s say there are ten infected monks. Most of the monks will see ten fellow monks with blue spots. Ten of the monks will see nine monks with blue spots. None of the monks will see only one monk with blue spots, or just two monks with blue spots, etcetera. Those monks seeing ten spotted monks have only two possibilities – either there are ten infected monks or there are eleven. Those monks seeing nine spots need only consider the possibility that there are either nine or ten monks infected. Most of the monks, in considering their first possibility (that there are only ten infections) realize that those ten, in considering the possibility that there are only nine infections, must allow for the possibility that there are nine monks who see only eight infected monks. With the information available, nobody sees any reason to consider any other lesser possibility. Just like in the conventional solution, each monk, having two possibilities, must allow the lesser possibility (which means they are not infected) to resolve itself first before concluding that they are infected and turn themselves in. All of the monks know that the least number of infections that any of them must consider is eight, not one. The first day nothing would happen. The second day, the monks that saw only nine other spotted monks will conclude that the possibility of anyone sighting eight spotted monks was groundless, so they will all turn themselves in. The third day, those who saw ten spotted monks will all breathe a sigh of relief.

Considering The Second Possibility

Now we need to look at the big picture. This isn’t rocket science. Why hasn’t anybody seen the error in conventional wisdom before today? I, like the monks, must consider the possibility that I am the one infected. If nobody else comes forward with a confirmation of the correct answer soon, I guess I’ll be forced to turn myself in. Please hurry.